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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | import OAI.Probability.ContinuumTransition.ConvexUniform
namespace OAI
universe u_1
section
open scoped Topology
open Set Filter
namespace ContinuumTemperature
lemma eventually_of_subseq (P : ℕ → Prop)
(h : ∀ φ : ℕ → ℕ, StrictMono φ → ∃ ψ : ℕ → ℕ, StrictMono ψ ∧ ∀ᶠ n in atTop, P (φ (ψ n))) :
∀ᶠ n in atTop, P n := by
by_contra hh
obtain ⟨φ,hφ,hbad⟩ := extraction_of_frequently_atTop (not_eventually.mp hh)
obtain ⟨ψ,_,hgood⟩ := h φ hφ
obtain ⟨n,hn⟩ := hgood.exists
exact hbad (ψ n) hn
lemma metric_tendsto_of_subseq {E : Type u_1} [PseudoMetricSpace E] {u : ℕ → E} {x : E}
(h : ∀ φ : ℕ → ℕ, StrictMono φ → ∃ ψ : ℕ → ℕ, StrictMono ψ ∧
Tendsto (fun n => u (φ (ψ n))) atTop (𝓝 x)) : Tendsto u atTop (𝓝 x) := by
apply Metric.tendsto_atTop.mpr
intro ε hε
have hh : ∀ᶠ n in atTop, dist (u n) x < ε := by
apply eventually_of_subseq
intro φ hφ
obtain ⟨ψ,hψ,ht⟩ := h φ hφ
exact ⟨ψ,hψ,ht.eventually (Metric.ball_mem_nhds x hε)⟩
exact eventually_atTop.mp hh
end ContinuumTemperature
end
end OAI
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